SATHEE: Matrices and Determinants 1 Question 11

Matrices and Determinants 1 Question 11

11. Let $X$ and $Y$ be two arbitrary, $3 \times 3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary, $3 \times 3$ , non-zero, symmetric matrix. Then, which of the following matrices is/are skew-symmetric?

(a) $Y^{3} Z^{4} - Z^{4} Y^{3}$

(b) $X^{44} + Y^{44}$

(d) $X^{23} + Y^{23}$

(2015 Adv.)

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Answer:

Correct Answer: 11. (c,d)

Solution:

Given, $X^{T} = - X, Y^{T} = - Y, Z^{T} = Z$

(a) Let $P = Y^{3} Z^{4} - Z^{4} Y^{3}$

Then, $P^{T} = {(Y^{3} Z^{4})}^{T} - {(Z^{4} Y^{3})}^{T}$

$= {(Z^{T})}^{4} {(Y^{T})}^{3} - {(Y^{T})}^{3} {(Z^{T})}^{4}$

$= - Z^{4} Y^{3} + Y^{3} Z^{4} = P$

$∴ P$ is symmetric matrix.

(b) Let $P = X^{44} + Y^{44}$

Then, $P^{T} = {(X^{T})}^{44} + {(Y^{T})}^{44}$

$= X^{44} + Y^{44} = P$

$∴ P$ is symmetric matrix.

Then, $P^{T} = {(X^{4} Z^{3})}^{T} - {(Z^{3} X^{4})}^{T}$

$= {(Z^{T})}^{3} {(X^{T})}^{4} - {(X^{T})}^{4} {(Z^{T})}^{3}$

$= Z^{3} X^{4} - X^{4} Z^{3} = - P$

$∴ P$ is skew-symmetric matrix.

(d) Let

$P = X^{23} + Y^{23}$

Then, $P^{T} = {(X^{T})}^{23} + {(Y^{T})}^{23}$

$= - X^{23} - Y^{23} = - P$

$∴ P$ is skew-symmetric matrix.

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Matrices and Determinants 1 Question 11

11. Let $X$ and $Y$ be two arbitrary, $3 \times 3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary, $3 \times 3$ , non-zero, symmetric matrix. Then, which of the following matrices is/are skew-symmetric?

Answer:

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एनसीईआरटी व्यायाम वीडियो समाधान

Matrices and Determinants 1 Question 11

11. Let X and Y be two arbitrary, 3×3, non-zero, skew-symmetric matrices and Z be an arbitrary, 3×3, non-zero, symmetric matrix. Then, which of the following matrices is/are skew-symmetric?

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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11. Let $X$ and $Y$ be two arbitrary, $3 \times 3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary, $3 \times 3$ , non-zero, symmetric matrix. Then, which of the following matrices is/are skew-symmetric?